NBER WORKING PAPER SERIES

CONDITIONAL MEAN-VARIANCE EFFICIENCYOF THE U.S. STOCK MARKET

Charles Engel

Jeffrey A. Frankel

Kenneth A. Froot

Anthony Rodriguez

Working Paper No. 2890

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138March 1989

We thank the Alfred P. Sloan Foundation and the Olin Foundation for generousresearch support. All errors are our own. This paper is part of NEER's

research program in Financial Markets and Monetary Economics. The viewsexpreesed here are those of the authors and do not necessarily reflect thoseof the Federsl Reserve Bank of New York, the Federel Reserve System, or theNational Bureau of Economic Research.

NBER Working Paper #2890March 1989

CONDITIONAL MEAN-VARIANCE EFFICIENCYOF THE U.S. STOCK MARKET

ABSTRACT

We apply the method of constrained asset share estimation (CASE) to test

the mean-variance efficiency (MVE) of the stock market. This method silows

conditional expected returns to vsry in unrestricted ways, given investor

preferences. We also allow conditionel verisnces to follow an ARCH process.

The data estimate reasonably the coefficient of relstive risk aversion,

though are unsble to reject investor riak neutrality. We reject the

restrictions implied by MVE, although chsnging conditional variances improve

statistically upon measured market efficiency. We find that unrestricted

asset-share and ARCH models help forecast excess returns. Once MVE is

imposed, however, this forecasting ability disappears.

Charles Engel Jeffrey A. FrankelDepartment of Economics Department of Economics

University of Virginia Harvard UniversityCharlottesville, VA 22901 Cambridge, MA 02138

Kenneth A. Froot Anthony RodriguesDepartment of Economics Federal Reserve Bank of New York

Massachusetts Institute of Technology Federal Reserve P.O. Station

Cambridge, MA 02139 . New York, NY 10045

Conditional Mean-Variance Efficiency

of the U.S. Stock Marhot

Charles Engel

University of Virginia and NEER

Jeffrey A. Franicel

Harvard University and NBER

Kenneth A. Froot

MIT and NBER

Anthony RodriguesFederal Reserve Bank of New York

This paper uses a technique that we call Constrained Asset Share Estimation (CASE) to test

the conditional mean-variance efficiency (MVE) of the U.S. stock market. The technique is useful

in time-series tests of simple asset pricing models because it allows estimated expected returns to

vary in an unrestricted way. It was first applied in a macroeconomic context in which the "market"

portfolio included not only equities, but also money, bonds and real estate.1 It has since been

applied more widely to other portfolios and has been extended to allow for variation in conditions!

second as well as first momenta, as in an autoregressive-conditional-heteroskedasticity (ARCH)

model.2

There is still a need for a clear statement of the advantages of the CASE method over earlier

tests of the MVE hypothesis for the stock market. Briefly, these advantages are of three sorts.

First, the technique does not impose the condition that expected returns are constant over time,

or even that they change in a slowly moving way. Rather it allows expected returns to vary

freely, as they must, for example, whenever new information which may not be observed hy the

econometricisn becomes availahle to the investor.3 In addition, in some of the tests below we allow

5ee Prankel (1982 W55a), Frankel and Dickens (1954), Frankel and Engel (1984)I sod w,us 19s2).°Feraon, Kandel and Stasnbaugh (1957) and Rayner (1998) test the constant-veriance vernon an stock portfnlica. Rodurtha

and Mark (19es), Bollerelev, Logic and woc,Idndge (1957) sod Eng.! and Rodrigu (1959), tat a vernon which allows forchanging conditional second moment- on portfolios, respectively, of stat, doniestk bonds, and short-term bills denominatedin differnit cunendes. --

In the tests below, expected excess returns are allowed to vary ins completely gensnl way as functions ofth. asset sharesrequiring only that a aet of prefeence pansnetere consistast with the Hare class of utility functions remain constant.

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second moments to vary according to an ARCH process.4 Allowing for such variation in conditional

moments is essential for a properly specified test of MVE. In fact, there is considerable evidence

that both the conditional expectation and conditional variance of excess returns contain important

predictable components.5

The second advantage of this method is that, by allowing the CAPM betas to evolve along

with the characteristics of the underlying assets, longer time series can be used to test MVE. In

the past, tests of unconditional MVE coped with changing conditional moments by using short

test periods, usually 5 years or less. There are two problems with this procedure. First, there

appears to be a substantial amount of conditional variation in both first and second moments over

forecast horizons of much less than 5 years.6 Second, while limiting time-series samples to 5 years

makes the assumption of constant conditional moments more believable, it also reduces the power

of tests of MVE. Low power can potentially explain the lack of any measured relationship between

risk and return in tests of MVE7 The use of longer time series also reduces the need to develop

small-sample test statistics, such as that suggested by Sbanken (1987). With large time-series

samples, the distributions of conventional test statistics are likely to be closer to their asymptotic

approximations.

The third advantage implicit in the CASE method is that-it nests MVE in a more general,

but economically meaningful, theory of portfolio determination. In contrast, most tests of the null

hypothesis of MVE have no clear alternative hypothesis. This feature is particularly important

because many tests do in fact reject MVE; when one rejects the null hypothesis it is crucial to

have some idea of what the alternative is. In some of the tests below, the alternative to MVE is

that investors' portfolio shares are linearly related to expected returns, and possibly to conditional

variances as well, but that investors do not compute covariances with the market portfolio in the

4The ARCH process does not allow second moments to vary frn& however. It is analogous to est,,nat,ng the first momentsby an ARIMA process, in which this period's expectation is reLated to recent eeal,sations, rather than by the CASE technique,in which expectations can vary freely

°See, for example, Fama and French (1955) and Potebs and Summers (1957) for evidence on the predictability of stockmarket returns, and Bollerilev (19s5) and Bollernier, Engis and Wooldridge 19s5) for evidence on the predictability of con-ditional variances of excess retums. These findings coupl with the results of Hansen and Richard (1957), who show that

the conditionally and unconditionally mean-variance efficient frontiers are ganerally different, nugget that audi variation in

conditional moments is important for tests of MVE.5Fama and French (19s5) document substantial mean reversion at forecast horisons of 5-5 yearn. Pindyck (1954) and Foterba

and Summers (tsse find evidence of high-frequency variation in conditional stock-market variances-TSee, fur example, Schwert (tSs3), Gibbons, Ross and Shanlsen (1955(, MacKinlay (19s7), and Gibbons and Shanken 10957)

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precise way that MVE would imply they should.5

Our tests below emphasize the nested nature of the hypotheses we consider. We pay special

attention to the importance of ARCH vs. MVE vs. the asset shares themsaives in explaining risk

premia. The broad findings can be summarized as follows. First, we find that stock-market shares

by themselves have statistically significant explanatory power in predicting monthly excess stock

returns. This is what we would expect if the etock market is mean-variance efficient and if required

returns change over time. However, we reject the restrictions implied by constant-variance MVE.

Moreover, the ability of asset shares to forecast future excess returns disappears once the MVE

restrictions are imposed. Something very different than MVE appear. to be responsible for asset

shares' ability to predict stock returns. Indeed, for a majority of the portfolios we construct, higher

conditionally expected returns are associated with lower value shares.

One might conjecture that MVE holds and that these results are an artifact of the maintained

assumption that conditional variances are constant. Indeed, we find that the data reject the hy-

pothesis that the market is mean-variance efficient with a constant variance against the alternative

that the market is mean-variance efficient with a conditional covariance matrix that evolves ac-

cording to an ARCH process. Time-varying second moments therefore move the mean-variance

efficient frontier closer to the market portfolio. This is good news for ARCH, but not for MVE: we

cannot reject the hypothesis that the ARCH-MVE model can explain no portion of excess returns.

Nevertheless, the data produce a sensible estimate of the coefficient of relative risk aversion of 2,

with a standard error of about 1.5, so that, while we cannot reject the hypothesis that investors

are risk neutral, we can reject hypotheses that they are strongly risk loving or risk averse.

Finally, we test a generalized ARCH specification, which does not impose MVE, against the

null hypothesis that the market is conditionally mean-variance efficient and that conditional vari-

ances evolve according to an ARCH process. Once again we reject the restrictions imposed by by

conditional MVE.

The paper is structured as follows. Sections 1 and 2 briefly describe tbe model and the data,

respectively. Section 3 tests for constant-variance MVE. We introduce our ARCH specification in

section 4, and test an unrestricted model as well as an ARCH-MVE system. Section 5 summarizes

tone possibility is that the managem of pension funds and the other funds that hold most equitis are concerned only withtninimieing the variance of their own performance, rtther thati computing cpearisnces with the aggregate portfolios held byindividuals as they in theory should.

3

our general nesting procedure for the hypotheses of interest and offers our conclusions-

1. The model

Mean-variance efficiency implies that the vector of conditional risk prernia is a linear combina-

tion of the asset shares in the portfolio1 with the weights proportional to the conditional variance

of asset returns:

= p11lAj, (1)

where E(r1+i) is the expected excess return above the riskless rate on an N x 1 vector of assets

conditional on all information available at time t, [Zg is the conditional variance of returns between

and t + 1, A is the N x 1 vector of portfolio weights, with E-A11= 1, and pj is a preference

parameter — the coefficient of relative risk aversion- If the aggregate stock portfolio is the "market"

portfolio, MVE is equivalent to the CAPM. To see this, note that the right-hand side of (1) ii

equivalent to the risk-adjusted conditional expected return on the aggreg&e (or market) portfolio.

Ei(ri÷j) = thEt(rnt+i) =

wberet11A cov(m1+i, ri+i)= —i--— = -

A1fl1A var(mj+i)

This expression makes it clear that the vector of sub-portfolio fits variee both with the shares o

assets in the portfolio, A1, and the conditional covariance matrix, 111, and thus may move substan

tially over short time intervals. Also, note that given preferences and n, (1) is a complete model a

expected excess returns: MVE implies that asset shares are sufficient statistics for optimal forecast

of excess returns.

Under rational expectations, we can replace the vector of expected excess returns with tli

actual returns by including a prediction error that is orthogonal to all information at time t:

= pgflAi + t+t,

where e — — Ei(ri+i). The insight in Frankel (1982) was that information about ti

conditional covariance matrix of returns can be obtained from the error terms since under MW

= E(1i4).4

MVE therefore imposes a set of restrictions that are highly nonlinear n hat they constitute pro-

portionality between the coefficient matrix and the veriance-covariance matrix of the error term in

(2).

To evaluate (3), we must take a position on whether li is constant over time, In sections 3 and

4 helow, we assume that fl is constant and that it follows an ARCH process, respectively. We test

the hypotheses that MVE holds against more general alternatives in which investors forecast excess

returns as a function of asset shares and past prediction errors. The exact specifications for the

alternative hypotheses are discussed in sections 3 and 4. We also test the MVE hypotheses above,

as well as the more general alternatives, against an even more restrictive null hypothesis; that

investors expect conditional excess returns to be zero. The results of these tests are also discussed

in sections 3 and 4. Section 5 presents a diagram which makes it easy to see the results of our

nested hypothesis tests.

2. The data

Our tests use monthly stock returns from the New York and American Stock Exchanges from

January 1955 to December 1984. Because of the computational difficulties in estimating (2) we

were forced to reduce the size of the cross section.9 In the tests below we aggregate stocks into

N = 11 (and sometimes 7) industry portfolios.

Table 1 describes the aggregation of stocks into industry portfolios. The returns for each

portfolio are value-weighted average returns. The N x 1 vector of portfolio shares, A1, is the value

of the stocks in the portfolios as a fraction of the total value of all stocks. Because it is desirable

to group together equities that have highly correlated returns, we ti'ied to put similar industries

into the seine portfolio.'° Stambaugh (1982) aggregates into 20 industries, roughly by type of final

output. We further aggregate into 11 industries, combining some of Stambaugh's catagories. Table

1 shows Stambaugh's 20 industries, as well as the 11-industry aggregation that we use to perform

our maximum likelihood tests of MVE. Table 1 also reports a 7-industry aggregation that we use

for the ARCH estimation in section 4.

°lf there are N assets, the computation involves a parameter matrix ot dimension N(N — 1)/In N(N — 11/2 thai must be

repeatedly inneted. Engel and Rodriguas (1988) offer a Wald teat version of the CASE test that gets around th,s problem, andallows one to consider larger vectore of assets. we apply it in Section 5 below.

LOOn the other hand, we would not waist to include together the suppliers of intermediate product. and the producers offinal output in the same industry, when steel prios rise, the cost of produc'oig autos increases so that it is poea.ble that steelproducers' profits rise when sum manufacturers profiti dccliii..

S

The value shares, A, are used to predict excess returns between time t and t + 1. The share

are measured monthly from the last day of January 1955 to the last day of November 1984 (S5

observations), while the returns are calculated as the dividend plus appreciation over the previou

month beginning the last day of February 1955 and ending the last day of December 1984. A

returns are nominal excess returns above the return on one-month Treasury bills recorded b

Ibbotson Associates (i986)

3. Tests of MVE with constant conditional variances.

If relative risk aversion and the return covariance matrix are constant, prflt p11, we ca

write demands for assets as a function of their own rate of return and returns on all other equitie

We would have,

A1

where B is an N x N matrix of coefficients. By inverting the system of equations in (4), we obta

a measure of expected excess returns,

= AAn (

where A = B'. This system of equations is a generalization of a static model of MYE. MS

imposes the restriction that the matrix of coefficients A be proportional to the variance of t

forecast error, Using cx post returns, (5) can be written:

= AA + t+I

Although the values of the equities are endogenous variables in an economic sense, they are si

uncorrelated with the prediction errors, which under rational expectations are uncorrelatcd w:

all information available at time t. (Under the null hypothesis that MVE holds precisely, prcdicti

errors are the only source of errors that enter the equation.11 ) Thus the system in (6) can

estimated consistently using ordinary least squares, equation by equation.12

Table 2 reports the results from estimating the unconstrained system of equations (6). Few

the coefficients individually are significantly different from zero. Unsurprisingly, the Rs are I

ttFor example, Enget end Rodrigue. (1959) uhow how jid meaaurement error in the ratee of return could be included in

residual t,+j.'2Note that the N need share,, A,.,. - are perfectly collinear because they sun to 1. This does not pose a problen

the estimation of 7), however, because the equations do not include a ronstant terns.

6

very high, and none exceeds .10. We can reject at the 95 percent tvti the hypothesis that the

asset shares have no explanatory power for excess stock returns. The log-likelihood value for the 11

equation system is 8709.35. The log-likelihood when all 121 coefficient are constrained to be zero is

8592.57. Twice the difference is distributed as y. The value of the statistic is 233.56 compared

with a critical value of 147.39.13 14

There is mixed support for one of our assumptions — that forecasts are rational. This assump-

tion implies that there is no serial correlation in forecast errors. We performed Breusch-Codfrey

tests for serial correlation from orders 1 to 20. We report the chi-square statistics only for the

tests of the existence of 20th order autoregressive or moving average errors. In only four of the

regressions can we reject the null hypothesis of no serial correlation up to 20th order at the 95

percent level.

Under the MVE hypothesie, this unconstraIned system of inverted asset demand equations is

not estimated efficiently. If we impose more structure on the system we can hope to improve the

precision of our parameter estimates. So we will estimate the system of equations in (6) imposing

the MVE constraints:

= $iX + q+i, (7)

so that A = p11. The N equation system (7) must be estimated y maximum likelihood techniques,

imposing an unusual cross-equation restriction — between the matrix of coefficients in the regressions

and the variance matrix of the regression errors. Note that the assumption that 11 is constant is

not the same as the usual assumption in MVE tests of constant betas and expected returns. As

we saw in the previous section, even with a constant covariance matrix, the betas, and hence the

expected returns on all securities including the aggregate or "market" portfolio, will vary over time

in a general, unrestricted way.15

Table 3 reports the maximum likelihijod results of (7). The log-likelihood value is necessarily

lower than the log-likelihood for (6) because (7) is a restricted form of (7): 8593.68 (as compared to

the unrestricted log-likelihood of 8709.35). We also report a chi-square statistic for the restrictions

The 99 percent critical value is 159.32."The only prior beliefs we have about the coefficients is that the return on asset j should be poutively related to the share

of asset jis, the total portfolio. If we thint of the market portfolio as comprised only of stocks, then in equilibrium tovestorowill demand a higher return from a given stock portfolio the more of it they are required to hold. Table 2 ehows that in 8 outof the It regreassona this own-coefficient ie negative (and significantly negative for induetñes 2 and 7). It is not eigvsiftcanttypositive in any of the regressions.

tFrant,et (1985e)

7

implied by (8). We impose 120 restrictions on the unconstrained system (121 coefficients are

constrained to be proportional to their corresponding elements in the variance matrix). The test

statistic is distributed Xso' and its value is 281.34. We can easily reject the hypothesis of MVE

at the 99 percent level. Comparing the results from table S to table 2, it is essy to see the source

of the rejection. When tbe coefficients are constrained, they are much smaller than when they are

unconstrained. Under the MVE constraints, an increase in the share of an asset has a much smaller

impact on risk premia.

If one were willing to accept the MVE estimates on the basis of prior beliefs, they yield in some

ways much more plausible asset pricing equations, We noted that in the unconstrained regressions

we frequently found that an increase in an asset share would actually decrease that asset's expected

return. That is not possible with the constrained MVII estimates.

Also, the point estimate of the coefficient of relative risk aversion, p, is very plausible — 2.03.

It is very close to the "Samuelson presumption" of a likely value for average risk aversion. The

coefficient is not estimated precisely, however, as it is not statistically different from zero at the 95

percent level. But its 95 percent confidence interval ranges only up to about 5.3 — still a believable

estimate for average risk aversion.

On the other hand, the constrained model does a very poor job of predicting excess returns.

The failure to reject the hypothesis that p = 0 implies that asset shares provide no statistically

significant explanatory power for risk premia under the MVII restrictions, because the coefficients

on the shares are all multiples of p. Above we mentioned that the log-likelihood when the coefficients

are all constrained to be zero is 8592.57. The likelihood under the MVII restrictions is only 8593.68

— a meager increase of 1.11. MVII vitiates the predictive power of the asset shares alone.

The estiriates reported in Tables 2 and 3 calculate the shares as a fraction of total equity

investment, if, however, there are positive net holdings of the riskless asset, then the shares should

properly be calculated as a fraction of total equity investment plus the total net value of the riskless

asset. The riskleas asset could have a positive net value if the government issues riskless short-tent

bonds, and investors consider government bonds to be additions to net wealth (so that they do not

fully discount future tax liabilities) or if the government issues money. We estimated the mode.

under the assumption that the relevant measore of the net supply is the value of all governmenl

S

bonds (which are calculated by Cox, 1985), and again under the assumption that the value of

outstanding Treasury bills measure the net supply of the riskiess asset. In both cases, there was

almost no change in the estimates.

We considered two other formulations for the coefficient of relative risk aversion, besides as-

euming that it is constant. In the first, we sssumed constant absolute risk aversion. In that case,

Pt = 514's where Ii is the coefficient of absolute risk aversion and Vf is the value of all equities at

time t. In the second, we considered a more general formulation consistent with the Hara class of

utility functions, Pt = + bW1. If Ii = 0 we have the constant relative risk aversion case, and if

a = 0 we have constant absolute risk aversion- Again, however, these versions of the model failed

to improve the constrained models performsnce)e

3.1. A wild test of MVE with constant conditional variances.

Maximum likelihood estimation of MVE is a difficult task. The constraints between the coef-

ficients and the variance cause grave problems in finding the maximum of the likelihood function.

The estimation is expensive and time consuming. The entire system must be estimated simultane-

ously, which in the case of the li-asset system means simultaneously estimating 122 coefficients.

The complexity of the problem increases with the square of the number of equations and assets.

if we were to estimate the model even for all 20 of Stanbaugh's original portfolios, it would mean

maximizing a very messy function over 401 parameters.

If we are interested in testing MVE, but not in actually obtaining the constrained coefficient

estimates, we do not need to estimate the constrained set of equations. A Wald test can be per-

formed using only the unrestricted model. In this case, the unconstrained model (6) is particularly

easy to estimate, because it requires only equation-by-equation ordinary least squares. Engel and

Rodrigues (1988) provide an expression for the Wild statistic for the MVE restrictions.

The WaId statistic is not difficult to compute even for large collections of assets. We can test

the MVE restrictions for the entire set of 20 industry portfolios composed by Stambaugh. We again

reject the MVE restrictions easily. The test statistic is distributed Xo, and has a value of 56.99,

well above the 99 percent level.17

151n arda to ewve space, we do not npofl these results."The comparable Waid test for the il-asset aggregation yielth a ststietc distributed as x, eqaa to 22.76. This also rejecta

the MVE restrictions at the 99 percent level These particuLar tests restrict only the diagonal elemenle or the return covarancematrix, and yet they reject easily.

9

The estimates of this section provide little support for MVE of the stock market. In all of

the tests performed, the restrictions that MVE places on a more general asset demand model are

strongly rejected.

4. Tests of MVE with ARCH conditional variances.

In the estimates reported in section 3, we assumed that return covariance matrix, O, was

constant over time. Because it has become clear in recent years that conditional variances show a

considerahle amount of variation, we turn to a model of time-varying conditional variances.

In simple regression models, the presence of heteroskedasticity often does not affect the con-

sistency of the coefficient estimates, although it does cause standard calculations of test statistics

to be inconsistent. When the MVE restrictions are imposed, however, changes in variances imply

changes in coefficient estimates, which in turn imply changes in expected excess returns. The coef-

ficient on the asset shares in the constrained model must move over time if U1 does, so holding U

constant leads to inconsistent coefficient estimates.

Inspection of (2) makes it easy to see why it is important to allow for variation in fl. There

are two possible sources of variation in expected returns if the measure of relative risk aversion

is constant: changes in asset shares, .X, and changes in fl. Suppose, for example, that favorable

news about a stock is announced. One could easily think of cases in which the price is pushed up,

increasing the stock's share in the aggregate portfolio, even though its expected return is now lower

with the news. If the market is mean.variance efficient, this can happen when the riskiness of the

asset declines — its own variance falls, or its variance with other assets declines. But, for the jth

asset, this is exactly a change in the jth row of fl.

The burgeoning econometric literature that proposes general corrections for heteroskedasticity

is not applicable to this model. That literature relies generally on procedures in which consistent

estimates of the residuals are obtained before any heteroskedssticity correction is made, and those

estimated residuals are used to construct beteroskedasticity-consistent statistics. In our MVE tests,

we must correct for time-varying variances when we estimate the regression coefficients because the

coefficients move with the variance. In order to do this, we need an explicit model of the variaocc

process.

Of course, our model is partial equilibrium in the sense that it does not indicate the nature o:

13

the exogenous variables that determine asset prices. It takes the stochastic processes of returns as

given, and computes the mean-variance efficient portfolio from these. In particular, it gives us no

indication of how variances should change over time.

We choose to model variances empirically following Engle's (1982) ARCH process. The ARCH

takes the conditional variance of this period's forecast error to be a function of past forecast errors.

It is not based on any theoretical notion of how the general equilibrium of the economy works. It

is an ad hoc model that seems to work well in practice.

The univariate representation of a first-order ARCH would be 4 = a+ 4, The variance

of the forecast error of the ith stock between time t and t+ 1 is given by or, and e is the

square of the forecast error made between time 2 — 1 and 2. This equation states that if we make

a large forecast error in one period, the variance of our forecast for the next period will he greater

(assuming y> 0).

In this section, we apply a multi-equation version of ARCH to the MVE problem. Because

of the difficulty in estimating large ARCH systems, we have further aggregated the assets into

the 7 portfolios described in table 1. Even with only 7 equations to estimate, the dimension of

the ARCH problem can be quite large. For example, even if we restrict ourselves to first-order

ARCH in which the variances and covariances this period are related only to the squares and cross-

products of forecast errors from the previous period, the problem is unmanageably large. There are

28 independent elements in the covariance matrix, If each element were linearly related to the 28

lagged squares and cross products of the forecast errors, there would be 812 variables to estimate.

More general forms of ARCH would relate the variance to more than one lag of the cross-products

of forecast errors, or to lagged variances (as in Bollerslev's (1986) GARCH).

Given the complexity of estimating the MVE-ARCH system, and given the limited amount of

data, it is helpful to lower the number of ARCH coefficients. Our teat of MVE uses a parsimonious

version of ARCH, in which the model,

Eg(r2i) = pflgA,

has return variance given by:

= PIP + G€t/1c.

U

We treat as parameters the upper triangular matrix F, and the diagonal matrix C. Under thi

formulation, each element of El1 is linearly related to its corresponding component in the matri

of cross-products of lagged forecast errors. There are only 35 coefficients to estimate. A furthe

advantage of the ARCH in (9) is that it enforces positive semi-definiteness on the covariance matri

fh. This turns out to be helpful in estimating the constrained model by maximum likelihood

The unrestricted form of the inverted system of asset demand equations is given by:

= A.A1. (c

MYB imposes the restriction that A1 = p(, where 1l is the conditional variance of r1+i. I

practice, if MVE is to be nested in the general system of asset demands, then the elements of A1 i

the general system might be related to the same variables that 121 is assumed to he related to, b

in an arbitrary way. More specifically, we assume that in the unrestricted model, the cnefficiei

matrix A1 evolves according to:

A1 = Q'Q + Fc'qF,

where Q is upper triangular and F is diagonal, and the conditional covariance matrix of return

fl, is given by (8). The MVE constraint, that A1 = pllg, imposes 34 additional constraints on ti

unconstrained asset demand equations in (9). —

For our restricted ARCH-MVE model in (8), the log-likelihood for observation is given by

= -.(7/2)ln(2r) — (1/2)l12t— (l/2)(rgi — P121A1)'QT'(rj+i

— pfl1A1), (1

where 111 is defined in (8), and e1 = r — p1l1_1A...j. Maximization of (if) is difficult for sevel

reasons, First is the constraint between coefficients and variances. Second is the recursive nato

of the problem (so that the likelihood at 2, defined above, depends on all observations from 1 to

Third is the large number of parameters to estimate simultaneously. We estimated the system

a modified version of a maximum likelihood program available in the Gauss programming packai

It uses a technique based on the Berndt, Hall, Hall and Hausman (1974) algorithm.

Before turning to the results of the ARCH estimation, it is useful first to examine the c

strained MVE estimates on the 7 equation system when 12 is constrained to be constant, as in I

previous section. Table 4 shows that the 7-equation system performs much like its 11-equation co

terpart. The estimate of the relative risk aversion parameter is close to 2.0. However, it is still

12

statistically different from zero, which indicates that the asset share data with the MVE constrai

imposed do a poor job of explaining expected returns The log-likelihood with MVE imposed

5558.56. This compares to a log-likelihood of 5603.56 for the corresponding constant-coefficie

unconstrained system of asset demand equations. In this case, MVE imposes 27 constraints on

general system. The test statistic is distributed with a size of 70.00. The MVE constrai

can be rejected strongly at the 99 percent level.

Table 5 reports the results of the MVE restrictions imposed on the ARCH system. There

two hypotheses to test here. The first asks whether we can reject the constant-variance MVE mo

in favor of the ARCH-MYE. A rejection would imply that time-varying variances statistically red'

the distance between the stock-market portfolio and the mean-variance efficient frontier. Suc

rejection would lead us to the other interesting hypothesis: can we reject the restrictions imp!

by MVE on the unrestricted ARCH system in (9) and (10)? This would involve a test of

hypothesis that Q = P and F = C.

The log-likelihood for the ARCH-MVE model in (9) is 5573.97. The constant variance vers

of MVE is a special case of this ARCH model, in which the G matrix from (9) is constrained tc

zero. This imposes 7 constraints on the ARCH system. Our test statistic is 30S2 and is distribu

we reject the constant-variance restrictions at the 99 percent level. ARCH therefore impn

significantly on the constant-variance form of MVE,

Four of the 7 ARCH coefficients (elements of the C matrix) are significantly different f

zero at the 95 percent level. These coefficients are all quite small in magnitude. The squat

each element gives the coefficient relating the variance in each equation to its own lagged squi

forecast error. Only one of the squared components of C is greater than .10.

The point estimate of p is 191 again close to the Samuelson value of 20. Once again

estimate is not statistically different from zero at the 95 percent level (although it is now signith

at the 80 percent level). The most important question is whether the ARCH-MVE model is

restrictive relative to the general ARCH system given in (9) and (io). This system will proc

a log-likelihood value at least as large as the value we reported above — 5603.56 — for the ver

of the unconstrained model in which A is constant. But even if its likelihood was no larger

this, the size of the test statistic (distributed Xs) for testing the MVE constraints on the Al

13

model would be SW 18. MVE would therefore be rejected at the 99 percent level. So we do not evea

need to estimate the unconstrained asset-pricing equations with A1 varying over time to know that

MVE is rejected.

We conclude that while letting the variance change over time is important in improving the

explanatory power of MVE, it does not improve it enough relative to an unconstrained system of

asset-demand equations.

14

5. Summary and Conclusions

Figure 1 provides a graphical sununary of our nested hypothesis tests. At the top of the figure

is the most unrestricted model we consider, the unrestricted ARCH model in equations (9) and

(10). At the bottom of the figure is the most restrictive model, that asset shares are of no help in

explaining required returns, or equivalently, that risk aversion is zero. For each pair of models, the

line connecting them reports the results of a test of whether the lower model (the null hypothesis)

can be rejected in favor of the upper model (the alternative hypothesis). It is easy to see that both

of the MVE formulations — the constant-variance case in equation (7) and the ARCH case in (S

— are rejected when compared with any more general alternative hypothesis. Worse, there is nc

evidence in favor of these MVE models even when they are pitted as alternative hypotheses agains'

the straw-man model in which asset shares don't matter at all (A1 0 in equation (9)).

There are several ways to rationalize these results. One would be that the true asset pricin:

model is not the CAPM, but rather the APT, a version of the intertemporal CAPM, or even th

one-period CAPM plus some other omitted varis.hle. A second explanation for the results woub

rely on the Roll (1977) critique. If the stock market is very unlike the true "market" portfolio, w

would not expect to find MVE, even if the CAPM holds.ts

Indeed, under this explanation, the asset shares and ARCH processes cannot be accuratel

observed. A third explanation of the results would be that the residuals in (2) lead to poor measure

of the conditional variances. If "peso problems" affect stock market returns, the estimated residual

will be biased. Imposing the MVE restrictions only compounds the problems. For example, it

well known that in the five years following the stock-market boom of August 1982, the market ro

at an average annual rate of 22 percent. Few would argue in retrospect that it is possible to obtai

from this period cx posi, valid measures of er eat e expected risk and return. Thus if the mod

in (8) were true we would expect that unconstrained asset shares and ARCH would predict exce

returns, but this could be erased by imposing the MVE restrictions which are not exactly satiafie

in our sample.

'45,milar reault. were tound, however, when money, bonds, and real estate were allowed into the portiolo (Frankel, 1955eand Frankel and Dickens, 1984) and when foreign assets were allowed (Frankel, 1952, and Frankel and Engel, 1984).

15

6. References

Berndt, E., R. Hail, B. Hail, and J. Hausman, 'Estimation and Inference in Nonlinear Structural Models," Annals of Economic end Social Measurement, 3 (1974), 653—55.

Bodurtha, James N., Jr., and Nelson C. Mark, 'Theting the CAPM with Time-Varying Risksand Returns,t Ohio State University, 1987.

Bolleralev, Tim, 'A Conditionally Heteroekedastic Time Series Model far Security Prices andRates of Return Data," Northwestern University, 1985.

Bollerslev, Tim, "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Ec-onornetrtcs, 31(1986), 307-327.

Bolierslev, T., HF. Engle, and .J.M. Wooldridge, "A Capital Asset Pricing Model with Time-Varying Covariances,' Journal of Political Economy, 96 (1988), 116-131.

Cox, WM., "The Behavior of Treasury Securities; Monthly 1942-1984," Research Paper no.8501, Dallas Federal Reserve Bank of Dallas, 1985.

Engel, Charles, and Anthony Rodrigues, 'A Wald Test of International CAPM," Universityof Virginia, 1988.

Engel, Charles, and Anthony Rodrigues, 'Teats of International CAPM with Time-VaryingConriances," forthcoming in Journal of Applied Econometrics, 1989.

Fama, Eugene F., and Kenneth H. French, "Permanent and Temporary Components of StockPrices," Journal of Political Economy, 96 (1988), 246-273.

Ferson, KR., S. Kandel, and HF. Stambaugh, "Teats of Asset Pricing with Time- VaryingExpected Risk Premiums and Market Betas," Journal of Finance, 42 (1987), 201-220.

Frankel, Jeffrey A., "In Search of the Exchange Risk Premium: A Six-Currency Test of Mean-Variance Efficiency," Journal of International Money and Finance, 2 (1982).

Fra.nkel, Jeffrey A., "Portfolio Shares as 'Bets-Breakers'," Journal of Portfolio Management,11 (1985a), 18-23.

Frankel, Jeffrey A., "Portfolio Crowding-out, Empirically Estimated," Quarterly Journal ofEconomics, 100 (iOSSh), 1041-1065.

Frankel, Jeffrey A., and William Dickens, "Are Asset-Demand Functions Determined byCAPM?," University of California, Berkeley, 1984.

Frankel, Jeffrey A., and Charles Eogel, "Do Asset Demand Functions Optimize Over theMean and Variance of Real Returns? A Six Currency Test," Journal of International

16

Economics, 17, December 1984.

French, Kenneth R. and Richard Roll, "Stock Return Variances," Journol of Financial Eco-nomics, 17 (1986), 5-26.

Gibbons, Michael R., and Jay Shanken, "Subperiod Aggregation and the Power of MultivariateTests of Portfolio Efficiency," Journal of Financial Economics, 19 (1987), 389-394.

Gibbons, M.R., S. Ross, and J. Shanken, "A Test of the Efficiency of a Given Portfolio,"Stanford University, 1986.

Hansen, Lars P., and Scott Richard, "The Role of Conditioning Information in DeducingTestable Restrictions Implied by Dynamic Asset Pricing Models," Econometrica, 55(1987), 587-614.

ibbotson Associates, Stocks, Bonds, Bills and Inflation: 1986 Yearbook, (lbhotson Associates:Chicago) 1986.

MacKinlay, Craig, "On Multivariate Tests of the CAPM," Journal of Financial Economics,18 (1987), 341-371.

Pindyck, R.S., "Risk, Inflation, and the Stock Market," American Economic Review, Vol. 74,pp. 335-51, June 1984.

-

Poterba, J.M. and L.H. Summers, "The Persistence of Volatility and Stock Market Fluctua-tions," American Economic Review, Vol. 76, pp. 11421151, 1986.

Poterba, James M., and Lawrence Summers, "Mean Reversion in Stock Returns: Evidenceand Implications," NBER Working Paper no. 1987.

Rayner, R,K., 1Generalized Instrumental Variables Estimation under Rational Expectationson First and Second Moments," Ohio State University, 1986.

Roll, Richard, "A Critique of the Asset Pricing Theory's Test.; Past I: On Past and PotentialTestability of the Theory," Journal of Financial Economics) 4 (1977), 129-76.

Schwert, C. William, "Size and Stock Returns, and othcr Empirical Regularities," Journal ofFinancial Economics, 12 (1983), 3-12.

Shanken, Jay, "Multivariate Tests of the Zero Beta CAPM," Journal of Financial Economics,14 (1985), 327-48.

Stsrnbsugh, R., "On the Exclusion of Assets from Tests of the Two Parameter Model: ASensitivity Analysis," Journal of Financial Economics, 10 (1982), 237-268.

Wills, H., "Inferring Expectations," London School of Economics, 1982.

17

Table I

industry Portfolios and S.E.C. Codes

S.E.C. Codes

1.. Mining 10, 11, 12, 13, 142. Food and Beverages 203. Textile and Apçare1 22, 23

4. Paper Products 265. Chemical 286, Petroleum 297. Stone, Clay and Glass 328. ,Prisary Metals 339. Fabricated Metals 3410. Machinery 35

Ii. Appiiarioes, Elec. Equip. 36

22. Transportat ion Equipment 3713. Misc. Manufacturing 38, 3914. Railroads 4015. Other Transportation 41, 42, 44, 45, 4716. Utilities 49

17. Department Stores 5318. Other Retail Trade 50—52, 54—5919. Bank., Fin., Real Estate 60-67

20. Miscelianenus 1,4,15—1121,24,25,27,30,3146,48,70,73,75,78,79,80,82,89,99

i ?ti° Combinations of the 20 portfolios)

Portfolio Industry Portfolios

1 1, 202 2, 3, 4

0

1 6

7, 8, 96 10

7 11

8 12—15

9 16

10 17, 18

11 19

1 cfckc Combinations of the 20 portfolios

Portfolio lndustry Portfolios

1. 2. 3, 4, 202 5, 7,8, 9

0

10. 11

12—15

16

17—19

Table 2

Estigated Coefficients from Unconstrained OILS 8egressions

Dependent Variable: Excess rate of return on asset 3Independent Variables: Shares of asset 3 in total portfolio

1 2 A k5 A8 i 110 X1'

Equation I—0.14 0.19 0.26 —0.06 —0.11 0.14 —0.70 0.08 0.21 —0.35 0.26(0.12) (0.82) (0.30) (0.26) (0.32) (0.25) 0.44) (0.22) 0.32) (0.25) (0.44)

82 .023 Breusch—Godfrey statistic (20 lagsi 42.79

-0.11 _2,29* 0,64* -0.29 -0.28 0.44 _1,12* 0.16 0.59* 0.83 2.06(0.13) (0,86) (0.32) (0,27) (0.34) (0.26) (0.46) (0.23) (0.22) (0.57) (1.24)

P2 .050 Breusch—Oodfrey statistic (20 lags) 23.38

qtipn I

-0.20 -1.05 0.12 -0.04 -0.32 0.14 _1.20* —0.02 0.46* 1.16 2.05

(0.13) (0.89) (0.33) (0.28) (0.35) (0.27) (0.47) (0.24) (0.23) (0.59) )1.29(

P2 .047 Breusch—Godfrey statistic (20 lags = 16.99

Equation 4

0.15 -0.55 0.74 _0.82* -0.81 0.14 -1.01 0.44 -0.01 -0.60 2.79(0.16) (1.09) (0.10) (0.34) (0.43) (0.33( (0.58) (0.29) (0.28) (0.72) (1.57)

P2 .027 Breusch—Uodfrey statistic (20 lags) = 21.74

Equation 5

-0.25 -1.00 0.83* —0.25 -0.81 0.18 _1.68* 0.50 0.41 —0.02 2.20

(0.16) (1.07) (0.39) (0.34) (0.42) 0.33) (0.57) (0.29) (0.28) (0.71) (1.55)

.044 Breusch-Uedfrey statistic (20 isgs 30.71

9I3rP0 c

—0.10 —0.19 0.46 —Li-IC —0.ri8 —0.15 —0.28 0.37 0.18 —0.06 1.99o.15 1.04) )0.38 (o.33 (0.41 0.32 (0.5h) (0.28) )0.27( (0.69) (l.51

.04h Breusch—Cedfrey statistic t20 lags) 20.41

Table 2 (continued)

)1 12 l 1 1 0 119PtAQ_n 1

—0.17 2.72 0.83* —0.26 0.71 0.44 _2.15* 0.37 0.75 1.21 3.15

(0.17) (1.13) (0.41) (0.36) (0.44) (0.35) (0.60) (0.30) (0.29) (0.75) (1.631

82 .066 Breusch-Godfrey statistic (20 lags) = 17.38

Equation 8

—0.14 —0.85 0.25 —0.10 —0.43 0.08 _1.41* —0.04 0.62 0.94 1.80

(0.14) (0,93) (0.34) (0.29) 0.36) (0.29) (0.49) (0.25) (0.24) (0.62) (1.34)

82 .067 Breusch-Godfrey statistic (20 lags) = 21.10

kuat ion 9

—0.09 —0.77 0.50 —0.10 —0.12 0.18 —0.64 —0.04 0.30 0.07 0.82

(0.12) (0.801 (0.30) (0.251 (0.31) (0.25) (0.431 (0.21) (0,21) (0.53) (1.16)

82 = ..•032 Breusch—Godfrey statistic (20 lags) 35,Q75

Equation 10

—0.11 —0.36 v.20 —0.10 —0.27 0.01 —0.56 0.06 0.41 —0.02 1.05

(0.161 (3.06) (0.39) (0.33) (0.42) (0.33) (0.56) (0.28) (0.28) (0.70) (7.53)

62 .02? Breusch—Godfrey statistic 120 lags) = 44.68*

U—0.04 0.25 0.13 0.19 0.09 0.24 0.13 0.19 0.54 —0.20 —1.31(0.14) (0.95) 0.35) (0.30) (0.37) (0.291 (0.50) (0.25) (0.25) (0.63) (1.37)

.027 - Breusch—Oodfrey statistic (20 lags) = 42.42*

sigyñfjcant at 95% level

(standard errnrs in parentheses

Table 3

CAFTI Estimation, constant 0, 11 assets

r+1 PCP1P)Lt +

Var{t+i)1*4 Log Likelihood —8593.684711 *1*

The estimate of the coefficient p:

2. 03 19

(L6130)

The estimate of the upper triangular untrix P:

.0398 .0322 .0334 .0385 .0411 .0346 .0404 .0331 .0257 .0316 .0374).OOl8)).OO2ll(.OO23I(.O028)I.0026)1.0026)l.0030)10025)1.0022.0029o1.00231

.0274 .0197 —.0033 .0166 .0198 .0223 .0189 .0089 .0252 .004?

LOOl1)(.OO15)).0O23)L0019.0025)1.0022)l.00191001QC2SH.OOlS1

.0204 .0042 .0044 .0097 .0097 .0078 —.0046 .0015 .0000(.0008)1 .0024)1.001511 .0019) (.0019)1 .0014)1.001811 .0018)) .0018)

.0360 —.0029 —.0019 —.0032 —.0017 .0018 —.0073 .0125(.0014)1 .0016)) .0021)1 .0019) (.0015)(.0018)(.0019)L0016)

.0276 .0058 .0102 .0090 —.0046 .0003 .0051).0011)1.0019H.0019)1.0013)1.O017)).°°17)).00164

.0304 .0068 .0050 —.0025 .0021 —.0009

.0011)1 .0016)) .00151) .0017)1 .0018)1.0014)

.0272 .0063 .0000 .0063 .0020I. 0011) 1 . 0014)) .0017 1) .001914 .0017

.0214 .0020 .0094 .0027(.0010)) .0018)) .0018)) .0017)

.0272 .0032 .0050

.0011)1.001?)) .0014]

.0287 .0006(.0013)) .0013

.0219

.0007

'5canar'; :rrI)7E it. oareratbeses -

*4*

Table 4

CAW Estimation, Constant 0, 7 assets

r+1 'IL +

Var((+1) VP

Log Likelihood = -5558.561247

The estimate of the coefficient p

2. 02778

tl.46639

The estimate

.03842

(.00150)

of the upper triangular matrix P:

.03935 .03711 .04015 .03640(.00185) (.00246) (.00213) (.00206)

.02075 —.00471 .01708 .01571(.00075) (.00225) (.00140) i.00149)

.03757 —.00296 —.00242).00115( (.00143) (.00120)

.02435 .00762(.00092) (.00137)

.02206

.00087)

*4*

.03782(.00197)

.00485(.00123)

.00140(.00117)

.00342(.00117)

.00735(.00108)

.00408(.00109)

.02034

(.00075)

.02695

.00190)

— .00389(.00153)

- .00033(.00166)

- .00136.00156)

.00269(.00149)

.02801

.00103)

i standar. errors to jrentheses)

.02700

1.00191)

— .00395(.00160)

.00084

(.00182)

— .00140(.00160)

.00253

.00158)

.02779

.00 109)

.03700

1.00200)

.00494

(.00130)

.00082

(.00128)

.00308

(.00122)

.00747

.00109)

.00391

.00112)

.0197 1

(.00082)

Table 5

CAPI Estimation. ARCH, 7 assets

r1 +

Var(E+1) P'P + GEtrt1G

Log Likelihood —5573.969787

The estimate of the coefficient (3:

1.91212(1.47685)

The estimate

03714

.00152)

of the upper triangular matrix P:

.03883 .03364 .04036 .03738

(.00189) (.00274) (.00213) (.00204)

.02050 —.00278 .01648 .01486

(.00077) 4.00233) 4,00150) (.00158)

.03541 —.00285 —.00127(.00116) (.00157) (.00124)

.02405 .00687

(.00095) (.00138)

.02118(.00096)

The estimates of the diagonal elements of 0:

.198l9 .13305 .31874 .06267 —.U3718 .15481 .17706

.03962 .04684) .u6162) ) .04517) .04355) (.09843) (.04668)

ujc;nrd ?rrors in parentheses)

Tests of the model•111 E, LE,.,E,.;I. C,.,

ARCH-MVE

A, .pfl,rnp(P'P+G E,E,G)

Unrsitrictsd, ARCH en.? Iicisnts

A, •QO.FE,E,F

invsttur risk *utrsIltyA, • C

Figure 1

Unr.stri.i.d C.n.tsnt coot helm?;

A, Q'O

u4. F.-,

Con.t.nt-vsri.nCt MVF

A, • p0 Pt

,.*c,